An official website of the United States government
Abstract To date, there is no consensus on the probability distribution of particle velocities during bedload transport, with some studies suggesting an exponential‐like distribution while others a Gaussian‐like distribution. Yet, the form of this distribution is key for the determination of sediment flux and the dispersion characteristics of tracers in rivers. Combining theoretical analysis of the Fokker‐Planck equation for particle motions, numerical simulations of the corresponding Langevin equation, and measurements of motion in high‐speed imagery from particle‐tracking experiments, we examine the statistics of bedload particle trajectories, revealing a two‐regime distance‐time (L‐Tp) scaling for the particle hops (measured from start to stop). We show that particles of short hop distances scale asL~giving rise to the Weibull‐like front of the hop distance distribution, while particles of long hop distances transition to a different scaling regime ofL~Tpleading to the exponential‐like tail of the hop distance distribution. By demonstrating that the predominance of mostly long hop particles results in a Gaussian‐like velocity distribution, while a mixture of both short and long hop distance particles leads to an exponential‐like velocity distribution, we argue that the form of the probability distribution of particle velocities can depend on the physical environment within which particle transport occurs, explaining and unifying disparate views on particle velocity statistics reported in the literature.
more »
« less
Analytical Solution for Anomalous Diffusion of Bedload Tracers Gradually Undergoing Burial
Abstract Accounting for the burial of tracer particles during bedload transport is an important component in the formulation of tracer dispersal in rivers. Herein we propose a modified active layer formulation, which accounts for the effect of burial and admits analytical solutions, enabling insightful exploration of the phenomenon of superdiffusion of bedload tracers at the intermediate timescale. This phenomenon has been observed in recent numerical results using the 2‐D Exner‐Based Master Equation. By assuming that tracers in the active layer can exchange with nontracer particles in the substrate layer to preserve mass, and that tracers entering the substrate layer get permanently trapped during the timescale of analysis, we are able to deduce governing equations for the tracer concentration in both layers. The active layer tracer concentration is shown to be governed by an advection‐diffusion equation with a sink term, and the increase of tracers in the substrate layer is driven by a corresponding source term. The solution for the variance of tracer population is analytically determined and can be approximated by the sum of a diffusion‐induced scaling (∝t1) and an advection‐induced scaling (∝t3) terms at the intermediate timescale, which explains the phenomenon of superdiffusion. The proposed formulation is shown to be able to capture the key characteristics of tracer transport as inferred by comparison with available results of numerical simulations.
more »
« less
- PAR ID:
- 10462928
- Publisher / Repository:
- DOI PREFIX: 10.1029
- Date Published:
- Journal Name:
- Journal of Geophysical Research: Earth Surface
- Volume:
- 124
- Issue:
- 1
- ISSN:
- 2169-9003
- Page Range / eLocation ID:
- p. 21-37
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract It has been suggested before that small-scale magnetic flux rope (SMFR) structures in the solar wind can temporarily trap energetic charged particles. We present the derivation of a new fractional Parker equation for energetic-particle interaction with SMFRs from our pitch-angle-dependent fractional diffusion-advection equation that can account for such trapping effects. The latter was derived previously in le Roux & Zank from the first principles starting with the standard focused transport equation. The new equation features anomalous advection and diffusion terms. It suggests that energetic-particle parallel transport occurs with a decaying efficiency of advection effects as parallel superdiffusion becomes more dominant at late times. Parallel superdiffusion can be linked back to underlying anomalous pitch-angle transport, which might be subdiffusive during interaction with quasi-helical coherent SMFRs. We apply the new equation to time-dependent superdiffusive shock acceleration at a parallel shock. The results show that the superdiffusive-shock-acceleration timescale is fractional, the net fractional differential particle flux is conserved across the shock ignoring particle injection at the shock, and the accelerated particle spectrum at the shock converges to the familiar power-law spectrum predicted by standard steady-state diffusive-shock-acceleration theory at late times. Upstream, as parallel superdiffusion progressively dominates the advection of energetic particles, their spatial distributions decay on spatial scales that grow with time. Furthermore, superdiffusive parallel shock acceleration is found to be less efficient if parallel anomalous diffusion is more superdiffusive, while perpendicular particle escape from the shock, thought to be subdiffusive during SMFR interaction, is reduced when increasingly subdiffusive.more » « less
-
Abstract We begin with a treatment of the Caputo time‐fractional diffusion equation, by using the Laplace transform, to obtain a Volterra integro‐differential equation. We derive and utilize a numerical scheme that is derived in parallel to the L1‐method for the time variable and a standard fourth‐order approximation in the spatial variable. The main method derived in this article has a rate of convergence ofO(kα + h4)foru(x,t) ∈ Cα([0,T];C6(Ω)),0 < α < 1, which improves previous regularity assumptions that requireC2[0,T]regularity in the time variable. We also present a novel alternative method for a first‐order approximation in time, under a regularity assumption ofu(x,t) ∈ C1([0,T];C6(Ω)), while exhibiting order of convergence slightly more thanO(k)in time. This allows for a much wider class of functions to be analyzed which was previously not possible under the L1‐method. We present numerical examples demonstrating these results and discuss future improvements and implications by using these techniques.more » « less
-
The cytoskeleton–a composite network of biopolymers, molecular motors, and associated binding proteins–is a paradigmatic example of active matter. Particle transport through the cytoskeleton can range from anomalous and heterogeneous subdiffusion to superdiffusion and advection. Yet, recapitulating and understanding these properties–ubiquitous to the cytoskeleton and other out-of-equilibrium soft matter systems–remains challenging. Here, we combine light sheet microscopy with differential dynamic microscopy and single-particle tracking to elucidate anomalous and advective transport in actomyosin-microtubule composites. We show that particles exhibit multi-mode transport that transitions from pronounced subdiffusion to superdiffusion at tunable crossover timescales. Surprisingly, while higher actomyosin content increases the range of timescales over which transport is superdiffusive, it also markedly increases the degree of subdiffusion at short timescales and generally slows transport. Corresponding displacement distributions display unique combinations of non-Gaussianity, asymmetry, and non-zero modes, indicative of directed advection coupled with caged diffusion and hopping. At larger spatiotemporal scales, particles in active composites exhibit superdiffusive dynamics with scaling exponents that are robust to changing actomyosin fractions, in contrast to normal, yet faster, diffusion in networks without actomyosin. Our specific results shed important new light on the interplay between non-equilibrium processes, crowding and heterogeneity in active cytoskeletal systems. More generally, our approach is broadly applicable to active matter systems to elucidate transport and dynamics across scales.more » « less
-
Abstract Predicting the transport of bedload tracer particles is a problem of significant theoretical and practical interest. Yet, little understanding exists for transport in rivers in the presence of bedforms, which may trap grains and thereby influence travel distance. In a series of flume experiments with a sandy gravel bed in a large experimental flume, bed elevation and tracer travel distances were measured at high resolution for a range of discharges. As discharge increased, bedform height increased and bedform length decreased, increasing bedform steepness. For all tracer sizes and flow conditions, bedforms act as primary controls on the tracer travel distances. Bedform trapping increases linearly with the ratio of bedform height to tracer grain size, with 50% trapping efficiency for a ratio of two and 90% trapping efficiency for a ratio of four. A theoretical model based on the extended active layer formulation for sediment transport is able to capture much of the distribution of measured travel distances for all tracer sizes and discharges, providing a first connection between tracer transport theory and bedform trapping and indicating normal diffusion of tracers at relatively small timescales. Variable bedform geometry can influence trap efficiency for individual bedforms and the theoretical model can help identify “preferential trapping” conditions. The distribution of tracer travel distances for a mixture of grain sizes and variable discharge, as expected in natural rivers, displays heavy tail characteristics.more » « less
